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Publication 17-CNA-024

Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace-Beltrami operator

Nicolas García Trillos
Brown University
Applied Mathematics
Providence, RI

Moritz Gerlach
Department of Mathematics and Computer Science
Saarland University

Matthias Hein
Department of Mathematics and Computer Science
Saarland University

Dejan Slepčev
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

Abstract: We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a $m$-dimensional submanifold ${\cal M}$ in $\mathbb{R}^d$ as the sample size $n$ increases and the neighborhood size $h$ tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of $O\Big(\big(\frac{\log n}{n}\big)^\frac{1}{2m}\Big)$ to the eigenvalues and eigenfunctions of the weighted Laplace-Beltrami operator of ${\cal M}$. No information on the submanifold ${\cal M}$ is needed in the construction of the graph or the "out-of-sample extension" of the eigenvectors. Of independent interest is a generalization of the rate of convergence of empirical measures on submanifolds in $\mathbb{R}^d$ in infinity transportation distance.

Get the paper in its entirety as  17-CNA-024.pdf

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